There is a working women's hostel in a town, where 75% are from neighbouring town. The rest all are from the same town. 48% of women who hail from the same town are graduates and 83% of the women who have come from the neighboring town are also graduates. Find the probability that a woman selected at random is a graduate from the same town

#### Solution

Let the total number of women be 100.

∴ n(S) = 100

Let event N: Women are from neighbouring town,

event W: Women are from same town and

event G: Women are graduates.

Number of women from neighbouring town,

n(N) = 75

Number of women from same town,

n(W) = 25

∴ P(N) = `("n"("N"))/("n"("S")) = 75/100` and

P(W) = `("n"("W"))/("n"("S")) = 25/100`

`"P"("G"/"N"), "P"("G"/"W")` represent probabilities that woman is graduate given that she is from neighbouring town or same town respectively.

∴ `"P"("G"/"N") = ("n"("G"/"N"))/("n"("S")) = 83/100` and

`"P"("G"/"W") = ("n"("G"/"W"))/("n"("S")) = 48/100`

By Bayes’ theorem, the probability that a women selected at random is a graduate from the same town, is given by

`"P"("W"/"G") = ("P"("W")"P"("G"/"W"))/("P"("W")*"P"("G"/"W") + "P"("N")*"P"("G"/"N"))``

= `((25/100)*(48/100))/((25/100)*(48/100) + (75/100)*(83/100))`

= `(25 xx 48)/((25 xx 48) + (75 xx 83))`

= `48/(48 + 249)`

= `16/99`